The formal study of the properties of logical systems as objects of investigation: their syntax, semantics, proof transformations, decidability, completeness, compactness, and other metalogical theorems that are stated about whole systems rather than individual formulas.

Principle

Treat logical calculi as formal objects and formulate general rules and limits about consequence relations, derivability, model theory, and the relationships between syntactic and semantic characterizations.

Demonstration

Proving the completeness theorem for first-order logic (showing every semantically valid formula is syntactically derivable) and deriving the undecidability of validity for first-order logic are typical metalogical demonstrations.

Misapplication

Confusing an object-level statement with a metalogical claim (for example, reading a metatheorem that quantifies over theories as if it were a theorem inside a fixed theory) or applying a model-theoretic result to a syntactic derivation without checking representability.

Consequence

Clarifies what can and cannot be proved inside given systems, allows transfer of results between logics, and yields principled limits (undecidability, incompleteness, completeness) that guide design and use of formal systems.

Reversal

Instead of reasoning about logics as objects, treat a logic solely as an object-level calculus whose only relevant facts are individual formula derivations and semantic evaluations within a fixed model.

Boundary

Focuses on formal properties of logical systems and excludes empirical, psychological, or informal philosophical accounts of reasoning; applies chiefly to formally specified calculi and their semantics, not to informal argumentation.

Semantic Tension

Tension arises between syntactic (proof-theoretic) and semantic (model-theoretic) approaches: some results are perspectival (provability vs validity) and translating between them requires representation and soundness assumptions.

Synthesis

Meta-logic coherently unites the study of syntax, semantics, and transformation rules to produce general theorems about entire logical systems, revealing both capabilities (completeness, decidability) and limitations (undecidability, incompleteness).